The growth of a dominant crack in a creeping material

Scripta

METALLURG!CA

V o l . 16, pp. i 0 9 - i 1 4 , 1982 P r i n t e d in t h e U . S . A .

Pergamon P r e s s Ltd. All rights reserved

THE GROWTH OF A DOMINANT CRACK IN A CREEPING MATERIAL

A.C.F. Cocks* and M.F. Ashby# *Department of Engineering, Leicester University, Leicester, England. tDepartment of Engineering, Trumpington Street, Cambridge CB2 IPZ, England.

(Received

November

I.

6,

1981)

Introduction

Over the past decade much effort has been made to correlate creep-crack growth rates with parameters which characterise the crack tip field (Ellison and Harper, 1978); but very little progress has been made in deriving growth laws from consideration of the microscopic processes that occur ahead of the crack tip. Here a simple theorem is given for the growth of a dominant crack in a creep-ductile material under condition of plane stress. The assumption of creep ductility means that elastic effects can be ignored. The strain rate ahead of the crack tip for a growing crack is given by Hutchinson (1968) and Rice and Rosengren (1968): n n = kC *n+l r n+l ~~ (e) e e 1 ~ ~ where

(1)

k = (ii---) n+l (_~)n+l n (7 o

for a creep law of the form: o n = (e)

e o

(2)

o

where ~e and o e are the (Von Mises) equivalent strain-rate and s~ress respectively. In is a function of n (values of which are given by Hutchinson 1968) and ~ (O) is a function of O normalised with respect to the maximum effective strain-rate. C* is the creep J-integral (Ellison and Harper, 1978). Equation (i) gives the strain-rate ahead of the crack tip for a damage free material. With time, voids nucleate and grow in the region ahead of the crack tip. The material becomes softer and the load is shed to mere distant regions. Throughout this process the strain-rate within the damaged region must be compatible with that in the surrounding material. While the dimensions of the damage zone remain small compared with the dimensions of the sample, it is a reasonable assumption that the strain-rate is still given by equation (I) throughout damege accumulation. This process can be best illustrated by referring to Fig. i. The bars are constrained to deform at a displacement rate ~ in the direction of the applied load P. Bar I suffers the greatest strain-rate, and (through equation 2) is the most highly loaded member; "damage" appears in it first and it sheds part of its load which is redistributed amongst the other bars. This additional load is small compared to the load that the bars already carry. The load points P will therefore continue moving at the same velocity ~, throughout dammge accumulation in bar i. The bar is therefore subjected to a constant displacement rate rather than constant load. In a similar way, the material ahead of a crack tip is forced to deform at a constant rate throughout damage accumulation. 109 0036-9?48/82/010109-06503.00/0 Copyright (c) 1 9 8 2 P e r g a m o n Press

Ltd.

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Creep Crack GrOwth Model

Consider a zone of extent ro ahead of a growing crack (Fig. 2). The dimension r o is chosen so that the strain that an element of material experiences while it is outside the zone is negligible compared to the strain that accumulates in it as it moves across the zone. In moving from the edge of the zone to the crack tip it experiences a total strain, ge, which we equate to the critical strain, ec, for failure. % For plane stress conditions, Ee (0) in equation (i) has a maximum value of unity on the plane containing the crack. Then, (from equation (I): n

n

= k C *n+l r

n+l

e Let time t = 0 after a time t

when the element of material it is a distance: r=r

ahead of the crack and is experiencing

is a distance

r = ro

ahead of the crack tip. Then

da t dt "

o

a straln-rate: n k C *n+l

= e

As the crack grows,

the strain accumulated .t=ro/~t

in the element is:

U =0 * n/n+l [ C* n/n+l k C k dU da t)n/n%l . . . . da/dt u

ee = 0

n da t)n¥i (r O - ~-{ •

(ro - d--{ "

U=r

o

If we now assume that the size of the zone in Fig. 2 is sufficiently small (ro << a) that the quantities da/dt, and C~ remain essentially constant as the element crosses the zone, we obtain: I • n/n+l rn+l k C (n + i) ee = da/dt o Setting

ee = ~c

gives: da _ dt

where

k

is given by equation

k C

• n/n+l

i I n+l (n + I) r O

~c

(3)

1

(i).

We now require a numerical value for strain to failure ec. If e c is a material property then it can be found from a test in unlaxial tension*. Alternatively we can think of the element of material, ahead of the crack tip, containing voids, and calculate the strain accumulated as they grow to the size at which they link. These two approaches will be dis cussed in turn. III. Using equation

(I), equation

Strain

to Failure as a Material Property

(3) can be written as:

n I * n _ ~ n+l da (__o_o)n+l fC ~n+l ro -= .~-. - (n + I) (4) dt o e o n c • It is anticipated that ec will depend on stress state (Cocks and Ashby, 1981). A simple tensile t e s t should give a value appropriate for a plane stress crack (as in thin sheet), but may overestimate that for a plane strain (as in thick plate).

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CRACK

GROWTH

Table

IN

CREEP

iii

I

Data for creep crack growth

TEMP oC

MATERIAL

.16 % C Steel " 316 Stainless " 304 Stainless

~o/O n (m2 kN_ 1)n hr_ 1

400 500 600 650 650

1.06 1.26 2.57 8.7 1.0

x x x x x

10 -74 10 -51 10-62 10-42 I0-51

n

I

Ec

r n

13.3 9.7 10.7 7.4 8.9

1.0 1.0 0.22 0.22 0.22

o

2.86 3.0 2.96 3.18 3.04

500 500 500 500 500

pro! p~ ~ ~ pml

Table 2 Predicted values of

TEMP Oc

MATERIAL .16 % C Steel " 316 Stainless " 304 Stainless

A

A, (m hr kN-1)nln+Im hr- 1

400 500 600 650 650

1.97 3.40 5.11 7.25 5.43

× x × × x

I0 -S IO-5 lO-5 iO-S lO-5

Taira et al. (1979) measured crack growth rates in thin tubular specimens (that is, in plane stress). Their results are shown in Figs. 3, 4 and 5 for the three materials they tested. Equation (4) is plotted as a solid line on these figures. Values of ~^/~o, n and Ec were given by Taira et al., from tests on uniaxial specimens. In was taken ~rom Hutchinson (1968). The length ro must be large compared to the characteristic length over which any microscopic mechanism operates~ (the grain size d, for instance): in each case it was set equal to 500 ~m. The values of these variables are given for each material in Table 1. Stress is measured in units of kN m -2 and time in units of hrs. Agreement between experiment and theory is as good as can be expected for measurements of this type. The crack growth law for each material

is:

da = A C* n/n+ 1 dt Values of A are given in Table 2. and C* in units of kN m/m 2 hr. IV.

(5)

The crack growth rate

Strain to Failure

da/dt,

is measured

m/hr

Calculated from a Model for Void Growth

A recent analysis of grain-boundary void growth under multiaxial law creep (Cocks and Ashby, 1980) gives the strain to failure as:

ec

in units of

stress states by power-

i I 6 (n + I) En ((n + i) fi ) + 0.2 ~

where f. is the initial area fraction of voids is their spacing. ~ is defined by:

(or of the inclusions

6 = sinh - {2 (n - II (n+ ~--} e

(6) from which

they grow) and

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where p is the hydrostatic component of the local stress field. (This result was derived for constant stress boundary conditions but is also valid for displacement boundary conditions provided the ratio p/o e remains constant and ro >> d). Hutchinson (1968) gives the ratio of the stress components ahead of the crack tip. Grainboundary sliding can further increase the triaxiality (Cocks and Ashby, 1981). Taking this into account B can be calculated: it ranges from 1.6 for n = 3 to 2.6 as n approaches infinity, If r o ~ d then the above method of analysis breaks down. But we can then analyse the growth of the crack in the following approximate way. Define the mean strain-rate of the material over one grain diameter as: d

L

=

I

n

n

k C *n+l r

n+l dr d

O n =

Under this strain-rate

(n + I) k C * n + l

the voids will grow and eventually L

where tance

e dc

n d

n+l

coalesce after a time

equation

tf

given by:

tf = E c

(8)

is given by equation (6). After the grain boundary So the crack growth rate is: da = d dt tf

Substituting

(7)

(7) and (8) into equation

fails the crack advances a dis(9)

(9) gives: n

1

d a = (n + i) k C *n+l dn+l dt e

(I0)

C

This is identical with equation (3) so we obtain the same growth law for ro >> d and for r o = d. We may then assume that equation (3) is a reasonable approximation for any value of r o. For large values of n the crack growth-rate is not very sensitive to the value of ro. We can therefore obtain a good estimate of the growth rate by setting (ro/d) I/n+l equal to unity. Using this, with equations(3) and (6), gives an explicit dependence of crack growth rate on the initial volume fraction and spacing of voids (or inclusions), and on the grain size. n da dt

1

(n + I) k C*n+l d n+l i I 0.2 (n + I) B ~n t k(n + I) f.) + d

(11)

l

Note that the crack growth rate increases

as

fi

increases,

V.

Discussion

and as the grain size increases.

The crack growth rate for a ductile material has been calculated assuming that the strainrate ahead of the growing crack is given by the HRR relationship (equation (i)) even when the material is damaged. The results compare satisfactorily with experiments. The approach can be generalised to include void growth by a coupling between power-law creep and diffusion, but the results cannot be expressed in closed form. In the limit where diffusive growth acts alone (Raj and Baik, 1980) elastic effects become important and the crack growth is controlled by the elastic stress intensity factor, K. The method described has much in comznon with that given recently by Riedel,(1980). But it differs in one important respect. Instead of using the strain-rate for a damage free material ahead of the crack tip (as we do), Riedel uses the stresses and assumes that the voids grow under the influence of these stresses. His results for the power-law creep mechanism are not

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very different from that derived here. But for diffusive and coupled mechanisms, approach will give substantially different results to his. VI.

113

the present

References

Cocks, A.C.F. and Ashby, M.F. (1980) Metal Sci., 14, 395. Cocks, A.C.F. and Ashby, M.F. (1981) Prog. Mat. Sci., to appear. Ellison, E.G. and Harper, M.P. (1978) Jnl. Strain Analysis, 13, 35. Hutchinson, J.W. (1968) J. Mech. Phys. Solids, 16, 13. Raj, R. and Baik, S. (1980) Metal Sci., 14, 385. Rice, J.R. and Rosengren, G.F. (1968) J. Mech. Phys. Solids, 16, p.l. Riedel, H. (1980) "3rd IUTAM Symposium Creep in Structures", ed. Ponter, A.R.S., Springer-Verlag, Berlin. Taira, S., Ohtani, R. and Kitamura, T. (1979) J. Eng. Mat. and Tech., i01, 154.

Fig. i. The creep zone at the tip of a crack, idealised as a set of bars deforming at a prescribed displacement rate.

I

~

o

dQ

\,

'~-

ro - - - - I

- ~.

~. / /

~ I-

A(IELERATFn

CTEP

Fig. 2. The zone of radius ro in which creep strains are appreciable.

114

CRACK GROWTH IN CREEP

o soo'c ,. AiR

16, No. I

/

A ~00"C IN VACUUM ~ i ~ "

~e 10

i

.~/#

Vol.

~

5~7x ~

16" j

104 I I0 102 CREEP J-INTEGRAL C~ kN,m/i~h

1o

Fig. 3. Creep crack growth in a O.16 % steel, predicted by eqn. 4, and compared with the data of Taira et al. (1979). 10"I

1~a

i o A

I 650°C

i

I ~ ~

I

' I

.

.1:

io

o

E

I

10

100

500

CREEP J INTEGRAL C '~" k l m l m i h

Fig. 4. Creep crack growth in a type 316 stainless steel, predicted by eqn. 4, and compared with the data of Taira et al. (1979).

/ ¢~o • 'o

o

o

°

o oo

,

' I/ /

I0"I

"•i

10"

I I

6S0 °C A IN VACUUM o IN AIR

o

30/+ SS

~1 ,I , I ,I 1°1o 1 lo m, ~" CREEPJ-INTEGRAL C~ kNm/m~h

Fig. 5. Creep crack growth in a type 304 stainless steel, predicted by eqn. 4, and co~ared with the data of Taira et al. (1979).